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On the Measurement of Qubits On the Measurement of Qubits (1) (2,3) Daniel F. V. James ∗, Paul G. Kwiat , William J. Munro(4,5) and Andrew G. White(2,4) (1) Theoretical Division T-4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. (2) Physics Division P-23, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. (3) Dept. of Physics, University of Illinois, Urbana-Champaign, Illinois 61801, USA. (4) Department of Physics, University of Queensland, Brisbane, Queensland 4072, AUSTRALIA (5) Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, UNITED KINGDOM To be submitted to Phys Rev A Date: February 1, 2008 Abstract We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum arXiv:quant-ph/0103121v1 21 Mar 2001 two-level systems (“qubits”). Our particular emphasis is on qubits realized by the two polarization de- grees of freedom of a pair of entangled photons generated in a down-conversion experiment; however the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the ad- vantage of producing density matrices which are always non-negative definite). In addition a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results. LA-UR-01-1143 ∗Corresponding author: Mail stop B-283, Los Alamos National Laboratory, Los Alamos NM 87545, USA; tel: (505) 667-5436; FAX: (505) 665-1931; e-mail: [email protected]. 1 I. INTRODUCTION The ability to create, manipulate and characterize quantum states is becoming an increasingly important area of physical research, with implications for areas of technology such as quantum computing, quantum cryptography and communications. With a series of measurements on a large enough number of identically prepared copies of an quantum system, one can infer,to a reasonable approximation, the quantum state of the system. Arguably, the first such experimental technique for determining the state of quantum system was devised by George Stokes in 1852 [1]. His famous four parameters allow an experimenter to determine uniquely the polarization state of a light beam. With the insight provided by nearly 150 years of progress in optical physics, we can consider coherent light beams to be an ensemble of two-level quantum mechanical systems, the two levels being the two polarization degrees of freedom of the photons; the Stokes parameters allow one to determine the density matrix describing this ensemble. More recently, experimental techniques for the measurement of the more subtle quantum properties of light have been the subject of intensive investigation (see ref. [2] for a comprehensive and erudite exposition of this subject). In various experimental circumstances is has been found reasonably straightforward to devise a simple linear tomographic technique in which the density matrix (or Wigner function) of a quantum state is found from a linear transformation of experimental data. However, there is one important drawback to this method, in that the recovered state might not correspond to a physical state because of experimental noise. For example, density matrices for any quantum state must be Hermitian, positive semi-definite matrices with unit trace. The tomographically measured matrices often fail to be positive semi-definite, especially when measuring low-entropy states. To avoid this problem the “maximum likelihood” tomographic approach to the estimation of quantum states has been developed [3–7]. In this approach the density matrix that is “mostly likely” to have produced a measured data set is determined by numerical optimization. In the past decade several groups have successfully employed tomographic techniques for the measurement of quantum mechanical systems. In 1990 Risley et al. at North Carolina State University reported the measurement of the density matrix for the nine sublevels of the n = 3 level of hydrogen atoms formed following collision between H+ ions and He atoms, in conditions of high symmetry which simplified the tomographic problem [8]. Since then, in 1993 Smithey et al. (University of Oregon), made a homodyne measurement of the Wigner function of a single mode of light [9]. Other explorations of the quantum states of single mode light fields have been made by Mlynek et al. (University of Konstanz, Germany) [10] and Bachor et al. (Australian National University) [11]. Other quantum systems whose density matrices have been investigated experimentally include the vibrations of molecules [12], the motion ions and atoms [13,14] and the internal angular momentum quantum state of the F = 4 ground state of a cesium atom [15]. The quantum states of multiple spin-half nuclei have been measured in the high-temperature regime using NMR techniques [16], albeit in systems of such high entropy that the creation of entangled states is necessarily precluded [17]. The measurement of the quantum state of entangled qubit pairs, realized using the polarization degrees of freedom of a pair of photons created in a parametric down-conversion experiment was reported by us recently [18]. In this paper we will examine techniques in detail for quantum state measurement as it applies to multiple correlated two-level quantum mechanical systems (or “qubits” in the terminology of quantum information). Our particular emphasis is qubits realized via the two polarization degrees of freedom of photons, data from which we use to illustrate our results. However, these techniques are readily applicable to other technologies proposed for creating entangled states of pairs of two-level systems. Because of the central importance of qubit systems to the emergent discipline of quantum computation, a thorough explanation of the techniques needed to characterize the qubit states will be of relevance to workers in the various diverse experimental fields currently under consideration for quantum computation technology [19]. This paper is organized as follows: In Section II we explore the analogy with the Stokes parameters, and how they lead naturally to a scheme for measurement of an arbitrary number of two-level systems. In Section III, we discuss the measurement of a pair of qubits in more detail, presenting the validity condition for an arbitrary measurement scheme and introducing the set of 16 measurements employed in our experiments. Section IV deals with our method for maximum likelihood reconstruction and in Section V we demonstrate how to calculate the errors in such measurements, and how these errors propagate to quantities calculated from the density matrix. II. THE STOKES PARAMETERS AND QUANTUM STATE TOMOGRAPHY As mentioned above, there is a direct analogy between the measurement of the polarization state of a light beam and the measurement of the density matrix of an ensemble of two-level quantum mechanical systems. Here we explore this analogy in more detail. 2 A. Single qubit tomography The Stokes parameters are defined from a set of four intensity measurements [20] : (i) with a filter that transmits 50% of the incident radiation, regardless of its polarization; (ii) with a polarizer that transmits only horizontally polarized light; (iii) with a polarizer that transmits only light polarized at 45o to the horizontal; and (iv) with a polarizer that transmits only right circularly polarized light. The number of photons counted by a detector, which is proportional to the classical intensity, in these four experiments are as follows: n = N ( H ρˆ H + V ρˆ V )= N ( R ρˆ R + L ρˆ L ) 0 2 h | | i h | | i 2 h | | i h | | i n = ( H ρˆ H ) 1 N h | | i = N ( R ρˆ R + R ρˆ L + L ρˆ R + L ρˆ L ) 2 h | | i h | | i h | | i h | | i n = D¯ ρˆ D¯ 2 N h | | i = N ( R ρˆ R + L ρˆ L i L ρˆ R + i R ρˆ L ) 2 h | | i h | | i− h | | i h | | i n = ( R ρˆ R ) 3 N h | | i (2.1) Here H , V , D¯ = ( H V ) /√2 = exp(iπ/4) ( R + i L ) /√2 and R = ( H i V ) /√2 are the kets rep- resenting| i qubits| i | polarizedi | i in − | thei linear horizontal, linea| i r vertical,| i linear| diagonali | i− (45o)| andi right-circular senses respectively,ρ ˆ is the (2 2) density matrix for the polarization degrees of the light (or, for a two-level quantum system) and is constant× dependent on the detector efficiency and light intensity. The Stokes parameters, which fully characterizeN the polarization state of the light, are then defined by 2n = ( R ρˆ R + L ρˆ L ) S0 ≡ 0 N h | | i h | | i 2 (n n )= ( R ρˆ L + L ρˆ R ) S1 ≡ 1 − 0 N h | | i h | | i 2 (n n )= i ( R ρˆ L L ρˆ R ) S2 ≡ 2 − 0 N h | | i − h | | i 2 (n n )= ( R ρˆ R L ρˆ L ) . S3 ≡ 3 − 0 N h | | i − h | | i (2.2) We can now relate the Stokes parameters to the density matrix ρˆ by the formula 3 1 i ρˆ = S σˆ , (2.3) 2 i i=0 0 X S whereσ ˆ = R R + L L is the single qubit identity operator andσ ˆ = R L + L R ,σ ˆ = i( L R R L and 0 | ih | | ih | 1 | ih | | ih | 2 | ih | − | ih | σˆ3 = R R L L are the Pauli spin operators. Thus the measurement of the Stokes parameters can be considered equivalent| ih to| − a | tomographicih | measurement of the density matrix of an ensemble of single qubits. B. Multiple beam Stokes parameters: multiple qubit tomography The generalization of the Stokes scheme to measure the state of multiple photon beams (or multiple qubits) is reasonably straightforward.
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